The debate over assessment of narratives will not be solved here, but what can be offered are some suggestions about things to consider assessing. For example what and how features of narratives could be assessed if any should be assessed at all.
We must be critical of the myriad of possibilities for assessment with narratives.

Some possibilities:

a non-marked journal

a peer review system

learner selects their favourites from a collection of entries and only the select few are assessed

diagnostic (at the beginning of the year, semester, unit, etc.)

There above suggestions are a few examples, however they may or may not work for various learners and situations and should ultimately be considered very carefully prior to use.